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Author: tgrothus

Not long ago, Jasper Johns, who is now 87 and widely regarded as America’s foremost living artist, was reminiscing about his childhood in small-town South Carolina. One day when he was in the second grade, a classmate named Lottie Lou Oswald misbehaved and was summoned to the front of the room. As the teacher reached for a wooden ruler and prepared to paddle her, Lottie Lou grabbed the ruler from the teacher’s hand and broke it in half. Her classmates were stunned.

“It was absolutely wonderful,” Mr. Johns told me, appearing to relish the memory of the girl’s defiance. A ruler, an instrument of the measured life, had become an accessory to rebellion.

I was walking a generic cubical die around a sheet of graph paper one day, contemplating mapping the surface of the die to the plane. Place a die on a square and assign the number from the top face of the die to that square. Then roll the die forward one square, right one square, back one square, and left one square onto the square on which you started. If you assign the numbers from the top face of the die to each square covered in turn, the number of the final square will be different than the number assigned to it at the start. To make the numbers come out consistently you have to rotate forward two squares, then right two squares, back two, then left two. By taking steps two at a time in a given direction, the numbers 1-6 can be distributed evenly and consistently across the plane. Only 3/4 of the squares in the plane are mapped to the faces of the cube in this manner and 1/4 are left blank.

The opposite sides of a die traditionally add up to seven, so the 1, 2 and 3 faces share a vertex. You can cut the edges and flatten out the 6-faced cube into a crucifix-shaped hexomino which can tile the plane by a series of rotations/translations to match the tumbling die pattern described above. First, rotate a copy of the hexomino 180° about the eccentric red point. (The rotated copy is shaded to make it clearly distinct.) This 2-hexomino cell can then be stepped across the plane in increments of 4 units, horizontally and vertically. Again, only 3/4 of the squares in the plane are mapped to the faces of the cube in this manner.

Note that walking a tetrahedral die on the plane can be done consistently such that the entire plane is covered, with no blank spaces.